Triangles without a common vertex

Let G be a graph with m edges, $t(G)$ triangles, and adjacency spectral radius λ(G)>√ m. Li, Feng and Peng (2025) conjectured that t(G)≥√ m-O(1) if no vertex belongs to every triangle of G. We prove the stronger bound t(G)≥λ(G)-C for an absolute constant C, and give an explicit choice of C. The proof bounds the sum of the squared adjacency eigenvalues other than the largest and smallest. Interlacing and matrix perturbation estimates then restrict the neighbourhood of a vertex corresponding to a largest entry of a Perron vector to two disjoint edges and isolated vertices. A weighted count of the triangles completes the proof.